Descriptional Complexity of Finite Automata -- Selected Highlights
For researchers in automata theory and formal languages, this survey highlights fundamental limits and open problems in descriptional complexity.
The paper surveys selected results on descriptional complexity of finite automata, including state complexity trade-offs between deterministic and nondeterministic automata, non-recursive trade-offs with context-free grammars, and uncomputability of state complexity for combined operations involving intersection and marked concatenation.
The state complexity, respectively, nondeterministic state complexity of a regular language $L$ is the number of states of the minimal deterministic, respectively, of a minimal nondeterministic finite automaton for $L$. Some of the most studied state complexity questions deal with size comparisons of nondeterministic finite automata of differing degree of ambiguity. More generally, if for a regular language we compare the size of description by a finite automaton and by a more powerful language definition mechanism, such as a context-free grammar, we encounter non-recursive trade-offs. Operational state complexity studies the state complexity of the language resulting from a regularity preserving operation as a function of the complexity of the argument languages. Determining the state complexity of combined operations is generally challenging and for general combinations of operations that include intersection and marked concatenation it is uncomputable.